Let S = {v₁ , v₂ , .... , v_k} be a basis for the subspace W of Rⁿ. Then a basis T for Rⁿ that contains S can be found by applying the method of Example 5 to the vectors

v₁ , v₂ , ...... , v_k , e₁ , e ₂, ...... , e_n.

Do this in Problems 17–20.

Find a basis T for R⁴ that contains the vectors v₁ = (3 , 2 , 3 , 3) and v₂ = (5 , 4 , 5 , 5).

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Example 5 Find a subset of the vectors v₁ = (1,-1, 2, 2), v2 = (-3,4, 1,-2), V3 = (0, 1,7, 4), and V4 = (-5, 7, 4,-2) that forms a basis for the subspace W of R4 spanned by these four vectors.